{"id":"d1ee05fa-44ad-4a15-b2ef-e22f925bc40b","arxiv_id":"2507.13900","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any connected linear algebraic group G over a number field, strong approximation with the Brauer-Manin obstruction holds for the classifying stack BG off any nonempty finite set of places.","lead":"A single-author paper proves a strong approximation theorem for classifying stacks of connected linear algebraic groups over number fields: local torsors can be approximated by a global torsor whenever a Brauer-Manin obstruction vanishes. It also builds a topological framework for adelic points of algebraic stacks and a Brauer-Manin pairing for stacks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.5 hinges on a non-quoted application of Borovoi–Demarche [5, Thm 6.1]; the exact hypotheses on S and on the stabilizer must be verified before the density claim follows.","rationale":"The reader's weakest assumption is the same as the load-bearing point I find: Theorem 5.5 is a conditional consequence of a theorem the paper paraphrases but does not state. My reading of the rest of the argument found no independent fatal flaw: the lift of a Brauer-orthogonal point to X(A^S)_{Br(X)} is supported by Proposition 5.3/Corollary 5.2 and Proposition 2.2; the topology is defined by a lifting presentation; and once a point t∈X(k), g∈SL(V)(k_S) with g·t∈f^{-1}(U) is obtained, the conclusion follows because f(t)=f(gt). The spectral sequence proof of Proposition 5.3 is terse, but Corollary 5.2 provides a more direct route to the same vanishing, so I do not treat that as the central risk. Thus the critical unverified link is the exact content of [5,6.1]. Since this was already flagged by the reader, the verdict remains CONDITIONAL; the concern would be settled by checking [5]. No change to the reader's verdict is warranted.","tokens_in":16278,"tokens_out":53738,"duration_ms":672477,"concrete_test":"Obtain Borovoi–Demarche, Comment. Math. Helv. 88 (2013), Theorem 6.1, and verify its hypotheses for X=G\\SL(V) with right H=SL(V)-action, G a connected linear algebraic group, and S as in Theorem 5.5. Specifically check: (i) the action convention and conclusion (does the theorem cover right actions and the density of X(k)H(k_S), or only left actions and a different set); (ii) whether the stabilizer is allowed to be any connected linear subgroup or must be reductive; (iii) the exact conditions on S (must it contain just one finite place plus all archimedean places, or more); (iv) any requirements on local/global points of X. If (i)–(iv) all pass, Theorem 5.5 is justified; if any fails, the proof of Theorem 5.5 has a gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.5 reduces strong approximation for BG to the equivariant statement that X(k)H(k_S) is dense in X(A^{S,k})_{Br(X)}, with X=G\\SL(V), H=SL(V) acting on the right, and invokes [5, Theorem 6.1] as a black box. The paper's paraphrase in §5.2 (“If S is a finite set of places containing such that strong approximation holds for H off S. Then [5,6.1] says…”) does not quote the theorem's precise hypotheses, and the proof never checks them. In particular, it is not documented whether [5,6.1] allows the stabilizer G to be an arbitrary connected linear group or requires additional hypotheses (reductivity of G, a condition on S beyond containing all archimedean and one finite place, or a specified action convention), nor is it documented that the theorem's conclusion is for the right action on G\\H rather than a left action on H/G. If any of these hidden conditions fails, the cited theorem cannot produce the required t and g, and the density result for BG is unsupported. The visible hypotheses—G connected, H=SL(V) simply connected semisimple, strong approximation for SL(V) off S, X(k) nonempty—appear satisfied, so this is a verification gap rather than a demonstrated counterexample.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a topology on the adelic points of a class of algebraic stacks over a number field, constructs a Brauer-Manin pairing for algebraic stacks, and proves a strong approximation theorem with Brauer-Manin obstruction for classifying stacks of connected linear algebraic groups. The main theorem (Theorem 5.5) asserts that if G is a connected linear algebraic group over a number field k and S is a finite set of places containing all archimedean places and at least one finite place, then the diagonal image of BG(k) is dense in BG(A^{S,k})_{Br(BG)}. The proof reduces the problem to an equivariant strong approximation theorem of Borovoi-Demarche applied to the homogeneous space G\\SL(V) under the right action of SL(V). The paper also proves a more general quotient-stack theorem (Theorem 5.4) and establishes foundational results on spreading out stacks, on the Brauer group of quotient stacks, and on the Brauer-Manin pairing for stacks.","tokens_in":16551,"tokens_out":27488,"duration_ms":289617,"significance":"If correct, the main theorem answers a concrete question: a finite collection of local G-torsors can be approximated by a single global G-torsor whenever the local data is orthogonal to the Brauer group of BG, making the Brauer-Manin obstruction the only obstruction to strong approximation for classifying stacks. The paper also contributes a stack-theoretic framework for adelic topologies, correcting an error in the topologization of adelic points of stacks in [7], and extends the Brauer-Manin formalism to Artin stacks. The proof of the main theorem is short and cleanly reduces to the deep external results of Kneser, Sansuc, and Borovoi-Demarche. However, the two load-bearing steps—the verification of the exact hypotheses of [5,6.1] and the spectral sequence proof of Proposition 5.3—are not fully documented as written, so the current version requires revision before the central claim can be considered established.","major_comments":[{"comment":"The proof of Theorem 5.5 invokes [5, Theorem 6.1] as a black box, but the paper does not quote the theorem's precise hypotheses and does not verify that they hold for X=G\\SL(V), H=SL(V), and the set S of Theorem 5.5. In particular, it is not documented whether [5,6.1] allows the stabilizer G to be an arbitrary connected linear algebraic group, whether S must satisfy any condition beyond containing all archimedean places and at least one finite place, whether X(k) must be nonempty, or whether the theorem's conclusion is stated for the right action of H on G\\H rather than the left action on H/G. Since this external theorem is the decisive step that produces the rational point t and the element g∈SL(V)(k_S), the density claim for BG is unsupported unless the author supplies the exact statement of [5,6.1] and checks each hypothesis explicitly. This is a load-bearing verification gap rather than a demonstrated counterexample.","section":"§5.2 and Theorem 5.5"},{"comment":"The proof of Proposition 5.3 is a compressed spectral sequence argument that is not fully checkable. The text asserts that the spectral sequence for the presentation X→[X/SL(V)] and the spectral sequence for the projection X×SL(V)→X are related by an automorphism of X×SL(V), and that Proposition 5.1 then gives the desired isomorphism, but it does not spell out the identification of the E_1-terms, the shift in indices, the convergence, or why the isomorphisms in H^0, H^1, H^2 from Proposition 5.1 suffice to identify the H^2_tors groups. Because Proposition 5.3 is used in Theorem 5.5 to transfer the Brauer-Manin condition from BG to X, this proof should be expanded, for example by using the Leray spectral sequence for the SL(V)-torsor X→[X/SL(V)] together with the vanishings of R^1 and R^2 f_* G_m established in Corollary 5.2.","section":"Proposition 5.3"}],"minor_comments":[{"comment":"The sentence 'If S is a finite set of places containing such that strong approximation holds for H off S' is missing words (presumably 'containing all archimedean places and such that'), and the phrase 'is dense is dense' contains a duplicate. Please correct both.","section":"§5.2"},{"comment":"The cross-reference 'Corollary 5.3' should be 'Proposition 5.3', and the reference to 'Proposition 2.2' for the triviality of SL(V)-torsors should be to Corollary 2.3.","section":"Theorem 5.5 proof"},{"comment":"There are several typographical errors: 'Knesser' should be 'Kneser', 'it's derived subgroup' should be 'its derived subgroup', 'diagrm' should be 'diagram', 'morhpism' should be 'morphism', and the notation for the off-S adeles appears inconsistently as 'A^{S,k}' and 'A S'; these should be unified.","section":"Throughout"},{"comment":"In the definition of the Brauer-Manin pairing for stacks, the sum is written over a finite set S, but the intended meaning is the sum over all v∈Ω_k, with S being a finite set containing the places where the given adelic point does not lift to O_v; the current wording is ambiguous and should be clarified.","section":"§4.3"},{"comment":"In the proof of Proposition 5.1, the phrase 'using the fact that SL(V) is equal to it's derived subgroup' should be corrected to 'its derived subgroup'; additionally, the citations to Sansuc's results could be made more precise by giving the specific statements used.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unverified application of [5,6.1] in Theorem 5.5. If the author can quote the theorem and confirm that its hypotheses are satisfied by X=G\\SL(V), the central claim is likely sound. The spectral sequence proof of Proposition 5.3 is also in need of expansion, but it is probably repairable. The paper makes a genuine contribution to the arithmetic of stacks, assuming these gaps are filled. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Speaking plainly: this is a real result, but the main theorem is only as solid as an unverified black box. The paper proves strong approximation with Brauer-Manin obstruction for BG for connected linear G over a number field, and that statement is not in the cited literature. The author deserves credit for the stack-theoretic infrastructure: a workable topology on adelic points of stacks that gets around the error in [7], a Brauer-Manin pairing for stacks, and the reduction of BG to G\\SL(V) via a faithful representation into SL(V). Proposition 5.3, the Brauer group isomorphism for the presentation, is a genuine piece of work even if the spectral sequence argument is compressed.\n\nThe soft spot is exactly what the stress-test flags. In §5.2 the paper paraphrases [5, Theorem 6.1] as \"If S is a finite set of places containing such that strong approximation holds for H off S. Then [5, 6.1] says...\" That sentence is garbled, and the theorem's actual hypotheses are never reproduced. The proof of Theorem 5.5 then applies the theorem to X=G\\SL(V), H=SL(V), without checking the conditions: is G allowed to be an arbitrary connected linear group, or does the theorem require reductivity? Is the action convention right? Are there extra conditions on S beyond containing all archimedean places and one finite place? The visible data—G connected, H simply connected semisimple, X(k) nonempty—suggests the application may well be valid, but a referee cannot certify it from the text. This is a verification gap, not a demonstrated counterexample. If the hypotheses fail in an unexpected way, the density claim for BG does not follow.\n\nThe paper is not circular: the central claim sits on Kneser, Sansuc, and Borovoi–Demarche, not on the author's own prior work. The one self-citation supplies a standard quotient-stack isomorphism and is not load-bearing. There are typos—\"is dense is dense,\" \"Knesser\"—and Proposition 5.3 could use a fuller derivation, but those are minor.\n\nBottom line: this paper deserves a serious referee. The question is natural, the infrastructure is useful, and the main theorem is plausible. But the author needs to state the quoted theorem's hypotheses and verify them for G\\SL(V). I'd send it out with a request for that revision.","headline":"A genuinely new strong approximation theorem for BG that hinges on an unverified black-box application of Borovoi–Demarche; send to referees with a request to check hypotheses.","tokens_in":17092,"tokens_out":4183,"would_cite":true,"duration_ms":46406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R34","14D23","14G05","14F22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Brauer-Manin obstruction is the only obstruction to strong approximation for classifying stacks of connected linear algebraic groups over number fields.","keywords":["classifying stacks","strong approximation","Brauer-Manin obstruction","algebraic stacks","G-torsors","number fields","homogeneous spaces","Brauer group"],"falsifier":"Find a connected linear algebraic group G and a finite set S of places (containing at least one finite place and all archimedean places) for which there are local G-torsors over the places of S that are orthogonal to the Brauer group of BG but cannot be approximated by any global G-torsor; this would refute Theorem 5.5. A more limited check is to verify the precise hypotheses of the equivariant strong approximation theorem cited in the proof on G\\SL(V) and to identify one connected G for which a hypothesis fails.","tokens_in":16053,"feed_emoji":"🧩","tokens_out":9040,"duration_ms":82698,"temperature":0.7,"pith_summary":"The paper asks when finitely many local G-torsors for a connected linear algebraic group G over the completions of a number field can be approximated by a single G-torsor defined over the field. Its answer is that the Brauer-Manin obstruction is the only obstruction: if the local torsors are orthogonal to the Brauer group of the classifying stack BG, then a global G-torsor exists that is simultaneously close to all of them. To get there, the paper builds a topology on the adelic points of algebraic stacks, constructs a Brauer-Manin pairing for stacks, and identifies the Brauer group of BG with that of a homogeneous space of SL(V). The final theorem then follows by applying an equivariant strong approximation theorem for homogeneous spaces. A sympathetic reader would take away that strong approximation with Brauer-Manin obstruction holds for classifying stacks of connected linear algebraic groups.","feed_headline":"Brauer-Manin obstruction alone blocks gluing local G-torsors","feed_subtitle":"A new theorem: local G-torsors can be glued globally when the Brauer-Manin obstruction vanishes.","key_machinery":"The argument is carried by a chain of reductions that ends at a known equivariant strong approximation result. First, since every connected linear algebraic group embeds into SL(V), the classifying stack BG is isomorphic to the quotient stack [G\\SL(V)/SL(V)] (Lemma 2.1). The paper proves (Proposition 5.3) that the pullback map identifies the Brauer group of this quotient stack with the Brauer group of the homogeneous space G\\SL(V), so the Brauer-Manin condition on BG becomes a condition on that space. At that point the proof invokes the equivariant strong approximation theorem for homogeneous spaces (cited as [5, Theorem 6.1]) to find a point of G\\SL(V)(k) up to an SL(V)(k_S)-translate inside any given open neighbourhood of the adelic point; its image in BG is k-rational. The remaining machinery is the topologization of adelic points of stacks and the construction of the Brauer-Manin pairing for stacks, which make the approximations and the obstruction well-defined.","core_discovery":"The central discovery is Theorem 5.5: let G be a connected linear algebraic group over a number field k, and let S be a finite set of places containing at least one finite place and all archimedean places. Then strong approximation for the classifying stack BG off S with respect to the Brauer group Br(BG) holds: the diagonal image of BG(k) is dense in the adelic points of BG off S that are orthogonal to Br(BG) under the Brauer-Manin pairing. In concrete terms, a finite collection of local G-torsors over the completions at places of S can be approximated arbitrarily well by a global G-torsor precisely when the collection satisfies the Brauer-Manin condition. This extends the classical Brauer-Manin theory from varieties and schemes to stacks.","pith_inferences":["The theorem suggests that for any quotient stack [Y/G] whose associated SL(V)-bundle Y ×_G SL(V) satisfies strong approximation with Brauer-Manin, the same should hold for [Y/G]; the paper proves this in a special case, and checking mild hypotheses on Y could extend it further.","One natural test is to ask whether the condition that S contains a finite place is necessary; the equivariant input may require it, and exploring the archimedean-only case could reveal a genuinely different obstruction.","Because the Brauer-Manin pairing on stacks is constructed via spreading out, it might also be defined for stacks over global function fields, where an analogue of the theorem would be a further test of the method."],"forward_implications":["For G = PGL_n, the theorem gives a stack-level statement of the classical fact that the Brauer group of the number field controls which local PGL_n-torsors can be approximated by a global one.","The Brauer-Manin pairing for stacks is a new tool that can be applied to other stacks of the form [Y/G], not just classifying stacks, whenever the reduction to a special-group quotient works.","The result converts the existence question for global G-torsors into a finite computation: evaluate the Brauer-Manin pairing on the given local torsors; if it vanishes, approximation is possible.","For quotients of groupic varieties (including certain toric quotients), the paper's Theorem 5.4 yields strong approximation with Brauer-Manin obstruction as well, giving a family of examples beyond classifying stacks."],"supporting_citations":[{"why":"Supplies the equivariant strong approximation theorem for homogeneous spaces that is the key density input in the proof of Theorem 5.5.","marker":"[5]"},{"why":"Provides the classical strong approximation theorem for SL(V) used to invoke the equivariant approximation theorem.","marker":"[16]"},{"why":"Also cited for strong approximation for SL(V); together with [16] it supplies the density of SL(V)(k) in the adelic points.","marker":"[23]"},{"why":"Gives the original idea for topologizing adelic points of algebraic stacks, which the paper adapts after repairing an error.","marker":"[7]"},{"why":"Supplies the topologization of adelic points on schemes and algebraic spaces on which the stack-level topology is built.","marker":"[9]"},{"why":"Establishes the Brauer group computations for linear algebraic groups used in Proposition 5.1 to prove the isomorphism of Brauer groups in Proposition 5.3.","marker":"[27]"},{"why":"Provides the reduction of quotient stacks to special-group quotients that underlies the main argument.","marker":"[2]"},{"why":"Also cited for Brauer groups of quotient stacks and the reduction to special groups, supporting the key structural step.","marker":"[12]"}],"fun_headline_variants":["Global G-torsors glue when Brauer-Manin vanishes","Brauer-Manin condition decides gluing local G-torsors","Brauer-Manin obstruction is only barrier to gluing torsors","Local G-torsors glue globally if Brauer-Manin vanishes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the cited equivariant strong approximation theorem applying to the specific homogeneous space G\\SL(V) under the action of SL(V) for every connected linear algebraic group G and every set S as in the theorem; if that theorem has additional hypotheses on S, stabilizers, or rational points that G\\SL(V) does not satisfy, the density claim for BG does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Global G-torsors glue when Brauer-Manin vanishes","Brauer-Manin condition decides gluing local G-torsors","Brauer-Manin obstruction is only barrier to gluing torsors","Local G-torsors glue globally if Brauer-Manin vanishes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000707,"raw_usage":{"total_tokens":3103,"prompt_tokens":782,"completion_tokens":2321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":2256}},"tokens_in":398,"tokens_out":2321,"duration_ms":17460,"temperature":1.0,"reasoning_tokens":2256,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:16:41.095684+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a connected linear algebraic group G and a finite set S of places (containing at least one finite place and all archimedean places) for which there are local G-torsors over the places of S that are orthogonal to the Brauer group of BG but cannot be approximated by any global G-torsor; this would refute Theorem 5.5. A more limited check is to verify the precise hypotheses of the equivariant strong approximation theorem cited in the proof on G\\SL(V) and to identify one connected G for which a hypothesis fails.","supporting_citations":[{"cited_title":"Manin obstruction to strong approximation for homo- geneous spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant strong approximation theorem for homogeneous spaces that is the key density input in the proof of Theorem 5.5."},{"cited_title":"Starke Approximation in algebraischen Gruppen. I","cited_arxiv_id":null,"evidence_quote":"Provides the classical strong approximation theorem for SL(V) used to invoke the equivariant approximation theorem."},{"cited_title":"The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups","cited_arxiv_id":null,"evidence_quote":"Also cited for strong approximation for SL(V); together with [16] it supplies the density of SL(V)(k) in the adelic points."},{"cited_title":"A Topology on Points on Stacks","cited_arxiv_id":"2005.10231","evidence_quote":"Gives the original idea for topologizing adelic points of algebraic stacks, which the paper adapts after repairing an error."},{"cited_title":"Groupe de Brauer et arithm´ etique des groupes alg´ ebriques lin´ eaires sur un corps de nombres","cited_arxiv_id":null,"evidence_quote":"Establishes the Brauer group computations for linear algebraic groups used in Proposition 5.1 to prove the isomorphism of Brauer groups in Proposition 5.3."},{"cited_title":"On the motivic class of the stack of bundles","cited_arxiv_id":null,"evidence_quote":"Provides the reduction of quotient stacks to special-group quotients that underlies the main argument."},{"cited_title":"Brauer groups and quotient stacks","cited_arxiv_id":null,"evidence_quote":"Also cited for Brauer groups of quotient stacks and the reduction to special groups, supporting the key structural step."}],"review_version":1}